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I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$.

A linked question could be: does there exist a Whitehead double with genus $1$?

Thanks to all!

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How about this? – HbCwiRoJDp Dec 29 '11 at 20:19
up vote 7 down vote accepted

Both the questions have affirmative answers. Moreover, it is true that the Whitehead double of every non-trivial knot is a knot with genus $1$.

In fact, if you consider the pattern $K'$ contained in the solid torus $D^2 \times S^1$, it is quite easy to construct a Seifert surface $S$ for $K'$ with genus $1$ and contained in the solid torus. Such a surface is built with two rings and one of them is twisted.

Added (Jim Conant): Here are a couple of pictures of the genus one Seifert surface for $Wh(K)$. enter image description here

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Thanks a lot, @Jim. The picture on the left is exactly what I had in mind, because I see it embedded in the tubular neighborhood of the companion. – Andrea Jan 4 '12 at 23:36
Thank you very much. – user21962 Jan 5 '12 at 9:37

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